Connes [2] showed that if M is an injective .sigma.-finite factor of type II $I_{1}$ and $B_{phi}$=C1 for some normal faithful state .phi. on M then M is isomorphic to the Araki-wood factor. In [7], Haagerup has succeeded to show that if M is an injective factor of type II $I_{1}$ with separable predual, then $B_{phi}$=C1 for every normal faithful state on M. Since injective factors of type II $I_{lambda}$, 0.leq..lambda.<1, were classified [9], this together classifies all injective factors of type III with separable predual. It is not known for non-injective case. In this paper we consider some conditions under which the bicentralizers be trivial.ivial.